The optimum design of a moving PM-type linear motor for resonance operating refrigerant compressor

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1 International Journal of Applied Electromagnetics and Mechanics 33 (2010) DOI /JAE IOS Press The optimum design of a moving PM-type linear motor for resonance operating refrigerant compressor Do-Kwan Hong, Byung-Chul Woo, Jae-Hak Choi and Dae-Hyun Koo Korea Electrotechnology Research Institute, 70 Boolmosangil, Changwon, , South Korea Abstract. Moving Permanent Magnet (PM) type Longitudinal Flux Linear Motors (LFLMs) are electromagnetic devices which can develop directly powerful linear motion. This paper deals with an effective analysis of PM-type LFLM by Finite Element Method (FEM) and optimum design for lightweight. This paper presents the statistical optimum design of PM-type LFLM to reduce the weight of a moving PM-type linear actuator with the constraint of avg. thrust force using the penalty method with characteristics function and kriging interpolation method. The contribution and effect of the each design variable on the characteristic function is evaluated by the analysis of means (ANOM) and optimum design set is determined. The generalized reduced gradient (GRG) algorithm is adopted in kriging model. Therefore, it is expected that the proposed optimization procedure using penalty method and kriging interpolation method can be easily utilized to solve the optimization problem with constraint of electric machine. Keywords: Penalty method, characteristic function, kriging interpolation method Analysis of Means (ANOM), statistical optimum design, moving Permanent Magnet (PM) type Longitudinal Flux Linear Motor (LFLM) 1. Introduction Linear motors are electromagnetic devices developing mechanical thrust without mechanical slidercrank system mechanism. The advantages of linear motors include low noise, reduced operating cost, and increased flexibility of operation because of the gearless feature [1]. Linear motor, however, have a few partial limitations. The major reason is the low power density due to the inherent large air gap. For the application of linear motors to high power system, therefore, Permanent Magnet (PM) type Longitudinal Flux Linear Motors (LFLMs) can be considered because these motors can develop high magnetic thrust and reluctance thrust in relatively small air gap. There are several partial examples of linear actuator and generator [2,3]. Figure 1 shows the structure of the developed PM-type LFLM for refrigerant linear compressor, linear actuator and electro-mechanical equivalent circuit. It has two important electromechanical components, the PM-type longitudinal flux linear actuator and resonant springs. By controlling the operating frequency of the linear type compressor around mechanical resonance frequency, it has higher efficiency than the conventional rotary type compressor. This paper presents the statistical optimum design of moving PM-type longitudinal flux linear actuator for refrigerant linear compressor using penalty method [4,5] and kriging interpolation method [6,7]. Corresponding author. Do-Kwan Hong, Tel.: , Fax: ; dkhong@keri.re.kr /10/$ IOS Press and the authors. All rights reserved

2 674 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor Fig. 1. Configuration of a moving PM-type LFLM for refrigerant linear compressor: (a) A resonant-type refrigerant compressor and (b) Linear actuator and electro-mechanical equivalent circuit. Generally optimization is used with 2 or 3 design variables. However, seven design variables using the table of mixed orthogonal array are utilized. Its design goal is to reduce the weight of a moving PM-type linear actuator with the constraint of avg. thrust force with respect to the initial machine. The effective design variables and their levels should be determined using table of mixed orthogonal array, Analysis of Means (ANOM) and pareto chart. The objectives of LFLM model are lightweight, small size, flat force at stroke and maximization of avg. thrust force. In this study, two statistical optimum design method were investigated to predict the weight of a moving PM-type linear actuator model: the penalty method with penalty function and the kriging interpolation method. The metamodels determined from the tow methods were compared with the true numerical model. finally, the most desired set is determined and the influence of each design variables on the objective function can be obtained. The ANSYS software was utilized for the numerical analysis of avg. thrust force. 2. Optimization and analysis method 2.1. Characteristic function with constraint using penalty method in discrete design space To solve a constraint problem, it is first needed to define a characteristic function. In penalty term method, the characteristic function includes the constraints by penalty function, which involves initial response values and the extent of influence of each constraint by penalty coefficient. The optimum design problem of LFLM to minimize weight with the constraints of thrust force can be expressed as the following equations; Minimize: W (dv 1,...,dv 7 ) (1) Subject to : F al / F max 1 0 (2) Where W (dv 1,...,dv 7 ) represents the weight of a moving PM-type linear actuator with the several design variables, and F al is an allowable thrust force. With the help of the penalty term method, penalty

3 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor 675 functions corresponding to each constraint is defined as follow; F (dv 1,...,dv 7 )=α Max[0, F al / F al 1] (3) Where α is penalty coefficient that determines the extent of influence of each constraint to the characteristic function. By combining the scaled objective function with the initial weight of a moving PM-type linear actuator, W 0 and penalty function, the characteristic function can be defined as follows; Ψ(dv 1,...,dv 7 ) new = W (dv 1,...,dv 7 )/W 0 + F (dv 1,...,dv 7 ) (4) 2.2. Kriging interpolation method in continuous design space Kriging is a method of interpolation named after a South African mining engineer D.G. Krige, who developed the technique while trying to increase accuracy in predicting ore reserves. In the kriging model, the global approximation model for a response y(x) is represented as y(x) =β + v(x) (5) where x is the design variable vector, β is a constant, and v(x) is the realization of a stochastic process. In Eq. (5), v(x) has the mean zero, variance σ 2, and non-zero covariance. The weight of a PM-type linear actuator W is replaced by y(x) to make a surrogate approximation model. Let ŷ(x) be an approximation model. Hereafter, ˆ means the estimator. When the mean squared error between y(x) and ŷ(x) is minimized, ŷ(x) becomes ŷ(x) = ˆβ + r T (x)r 1 (y ˆβq) (6) where r is the correlation vector, R is the correlation matrix, y is the observed data and q is the unit vector. The definitions of R and r are well explained in Refs [6,7]. The unknown correlation parameters of θ 1, θ 2,..., θ n defined in R are calculated from the formulation according to maximize [n s ln(ˆσ 2 )+ln R ] (7) 2 where θ i (i = 1,2,..., n) > 0. In this study, GRG (generalized reduced gradient) algorithm built in the EXCEL program was utilized to determine the optimum parameters Comparison of simulation with experiment for verification The comparison of the static force obtained from FEM simulation and experimental test is shown in Fig. 2 with good agreement. From the results, the use of FEM is validated as it can be observed. Figure 3 shows the stator and the mover(8 segment PM) for reference model, a moving PM-type LFLM prototype which is analyzed. 3. Optimization of a moving PM-type linear actuator 3.1. Design variables and levels Selection of the design variables is very important setup in optimization procedure. Seven dimensions are selected as the design variables as shown in Fig. 4. Table 1 shows the design variables and levels. Avg. thrust force and weight of experiment result according to change of the design variables are shown in Table 2.

4 676 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor Table 1 Design variable and level Level D. V dv dv dv dv dv dv dv D. V. : design variable. Fig. 2. Comparison of FEM results with experiments (reference model) according to MMF Characteristic function and ANOM Fig. 3. Prototype of reference model (inner, outer core and PM). Table 2 shows the table of mixed orthogonal array and simulation result by FEM. Table 2 represents the table of mixed orthogonal array, which is determined by considering the number of the design variables and each level of them. For the solution of a constraint problem, it is required to prepare a

5 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor 677 Table 2 Table of mixed orthogonal array L 18 ( ) Exp. dv 1 dv 2 dv 3 dv 4 dv 5 dv 6 dv 7 Avg. Weight Calculated thrust (kg) characteristic force(n) function Fig. 4. Design variables of PM-type linear actuator. characteristic function by which the feasibility of constraints can be evaluated. In Table 2, the values of the characteristic function are calculated based on the experiment. Still, desired optimal set cannot be selected with this value of characteristic function. The effect of each design variable and contribution on the characteristic function should be determined as shown in Fig. 5(c) by ANOM Kriging interpolation method The purpose of this paper is to minimize the objective function (W ) with constraints of avg. thrust force (F ). The kriging interpolation method, the optimum parameters of θ 1,..., θ 7 are determined by solving Eq. (7). Then, the estimator β calculated. The optimum estimators are shown as Table 3.

6 678 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor Table 3 Optimum β and correlation parameters for kriging models(n s = 100) Response Correlation parameter (corresponding design variable) θ 1 (dv 1) θ 2(dv 2) θ 3(dv 3) θ 4(dv 4) θ 5(dv 5) θ 6(dv 6) θ 7(dv 7) β Weight (W ) Avg. thrust force (F ) Fig. 5. Main effects for means of avg. thrust force, weight and characteristic function: (a) Avg. thrust force, (b) Weight and (c) Calculated characteristic function. The optimization formulation of a moving PM-type linear actuator can be defined as follows: Minimize : W (dv 1,...,dv 7 ) subject to: F al / F max dv , 4.23 dv , 1.8 dv 3 2.2, 1.8 dv 4 2.2, 2.7 dv 5 3.3, 1.17 dv , 12.6 dv (8) Where W (dv 1,...,dv 7 ) is the weight of a moving PM-type linear actuator, F max is the maximum avg. thrust force at stroke and F al is the allowable avg. thrust force considering safety factor 1 which was obtained by repeated experiments. The design process of a moving PM-type linear actuator using the kriging interpolation method is as follows: Step 1: DOE strategy First, sample points should be set up to obtain the metamodels of weight of a moving PM-type linear actuator and avg. thrust force. DOE strategies can be used to sample the design space. In this study, the table of orthogonal array is introduced to sample the design space.

7 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor 679 Table 4 Optimum solution Design variable Level dv 1 dv 2 dv 3 dv 4 dv 5 dv 6 dv 7 Initial size Optimum size(kriging) Optimum size (Characteristic function) Table 5 Comparison of initial and Optimum model Model Weight (kg) Avg. thrust force stroke 3 mm (peak-peak) Initial Optimum Characteristic function Ŵ ˆF Kriging (predicted) FEM W F Error (Kriging vs FEM) Variation between initial And optimum FEM % Step 2: Matrix experiment The responses of the weight and avg. thrust force are calculated for each row of matrix experiment. The number of experiments is identical to the number of rows in the matrix; that is, and experiment means on magnetic field analysis using FEM. The computation time for magnetic field analysis in 1 case(stroke 3 mm, step 0.5 mm) is about 1 hour 20 min (on the workstation using 8 CPUs and 32 GB main memory). ANSYS is used to solve. Step 3: Building and validation of metamodels Based on the response calculated from Step 2, the penalty method with characteristic function and the kriging models are constructed. To assess the metamodels, the error in the surrogate model is characterized by using a few metrics. Step 4: Calculation of optimum Once an approximate formulation for optimization is obtained based on the metamodels, any optimization method can be used. Since all the true functions of the optimization formulation are replaced by simple mathematical expressions, the computational cost of the optimization process is very low. In this study, the GRG algorithm is added. The kriging model is slightly better than the characteristic function with constraint. The optimum solution and result comparison of the optimization problem with constraint are respectively shown in Tables 4 and 5, where the ˆ means the predicted value and FEM is simulation result. Each optimum value of an approximate model was calculated by the GRG algorithm in EXCEL. Particularly, the developed EXCEL-Kriging program computed the optimum solution of kriging approximated model. 4. Conclusion This research deals with the optimization of a moving PM-type linear actuator reduce its weight and to improve avg. thrust force as constraint of the initially designed a moving PM-type linear actuator

8 680 D.-K. Hong et al. / The optimum design of a moving PM-type linear motor with many design variables. Statistical optimum design procedure included the penalty method with characteristic function in discrete design space to and kriging interpolation method in continuous design space. The most effective design variables are extracted by ANOM of avg. thrust force, weight and calculated characteristic function. Kriging interpolation method is better than penalty method result. The predicted performances of the optimum designed LFLM by kriging interpolation method are also in good agreements with the simulation results by FEM simulation for verification. Therefore, when this proposed approach is applied, it can efficiently raise the precision of the optimization and reduce the number of experiments in the optimization design. References [1] H. Lee, S.S. Jeong, C.W. Lee and H.K. Lee, Linear compressor for air-conditioner, Proc. of the 17th International Compressor Engineering Conference at Purdue, USA, 2004, pp [2] P. Zheng, A. Chen, P. Thelin, W.M. Arshad and C. Sadaranani, Research on a tubular longitudinal flux PM linear generator used for free-piston energy converter, IEEE Trans Magn 43(1) (January 2007), [3] D.K. Hong, B.C. Woo, D.H. Koo and D.H. Kang, Optimum design of transverse flux linear motor for weight reduction and improvement thrust force using response surface methodology, IEEE Trans Magn 44(11) (November 2008), [4] D.K. Hong, S.C. Choi and C.W. Ahn, Robust optimization design of overhead crane with constraint using the characteristic functions, International Journal of Precision Engineering and Manufacturing 7(2) (April 2006), [5] D.K. Hong, B.C. Woo and D.H. Kang, Optimum design of TFLM with constraints for weight reduction using characteristic function, IEEE Trans Magn 43(4) (April 2007), [6] K.H. Lee and D.H. Kang, Structural optimization of an automotive door using the kriging interpolation method, Proc. of the Institution of Mechanical Engineers, Part D: J. Automobile Eng 221(12) (2007), [7] A. Guinta and L. Watson, A comparison of approximation modeling techniques: polynomial versus interpolating models, Proc. of the 7thAIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, MO, AIAA, September 1998, AIAA , pp